LIDS-P-1186
March, 1982
A NEW BEHAVIORAL PRINCIPLE FOR
URBAN TRANSPORTATION NETWORKS
by
Stanley B. Gershwin
David M. Orlicki
This research was carried out in the MIT Laboratory
for Information and Decision Systems with support
extended by the U. S, Department of Transportation
under Contract DOT-TSC-RSPA-81-9.
Laboratory for Information
and Decision Systems
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
ABSTRACT
A new hypothesis on traveler behavior in a network is described which
is based on empirical findings in the theory of travel budgets.
It is
translated into an assignment principle for characterizing the distribution
of travelers, as well as the demand and mode split. A numerical technique
is proposed, and it is applied to several examples to illustrate qualitative
features.
The new hypothesis is significant because it considers all travel
decisions--whether or not to travel, where to go, and what mode to use-in a single, unified way. The feedback from travel time and money costs
on links and modes to traveler behavior is explicitly considered.
ACKNOWLEDGEMENT
We are grateful to the U. S. Department of Transportation, Research
and Special Programs Administration, and Transportation Systems Center
for supporting this work. We are particularly grateful for the help and
support of Robert Crosby and Diarmuid O'Mathuna.
Dr. Yacov Zahavi of
Mobility Systems, Inc. (Bethesda, Maryland) provided guidance in travel
budget theory and helped devise the large network. Professor Loren K.
Platzman of Georgia Institute of Technolgy was instrumental in helping
to create the equilibrium formulation.
1.
INTRODUCTION
1.1. Purpose
The purpose of this paper is to introduce a new model of traveler
behavior in a transportation network.
This model incorporates a mechanism
which treats route, mode and destination choices, and the decision of
whether or not to travel, in a single, unified formulation.
Models and computation techniques have been studied
(Dafermos, 1974;
Dafermos, 1980; Florian and Nguyen, 1974; Leblanc and Abdulaal, 1980)
which treat these issues in an integrated way.
However the models of
individual decisions are based on different assumptions.
route choice is often based on the logit model.)
(For example,
Some of the choice models
(such as the logit models) appear to be convenient methods of summarizing
local survey data rather than representations of universal human behavioral
mechanisms.
We use the term "behavioral principle" to be a set of statements that
characterize the choices made by individuals in a transportation system.
We use "assignment principle" to mean a set of statements that characterize
flows of vehicles or travelers in a system.
One of the purposes of this
paper is to emphasize the relationship between the two.
An assignment
principle is limited in its validity and predictive power unless it is
based on a single, unified behavioral principle.
The models presented in here are based on the following behavioral principle.
Travelers have maximum amounts of time and money that they are willing
and able to spend traveling during a day.
They may spend less but not more.
Within these constraints, they maximize the benefit they obtain from traveling.
This benefit is determined by the links and nodes of the network a traveler
passes and on the mode he employs.
An important concept is that of a journey.
journeys at the same point:
possibly with several loops.
a residence.
Travelers start and end their
They travel in a continuous path,
This differs from existing models, in which the
fundamental unit of travel is a trip.
1.2
Outline
Section 2 of this paper derives two versions of an assignment principle
from the behavioral principle mentioned above.
sented in Section 3.
A numerical technique is pre-
Sections 4 and 5 contain a set of examples.
The small
examples in Section 4 are intended to help explain the behavioral and assignment
principles; the large example of Section 5 demonstrates their realism and
applicability.
Further research topics are suggested in Section 6.
2
2.
BEHAVIORAL AND ASSIGNMENT PRINCIPLES
Traffic Assignment is the computation of vehicle and traveler flows in
a transportation network.
It is based on data on the travelers such as their
origins, possible destinations, and car ownership;
and on the physical attri-
butes of the network such as its structure and its link flow capacities.
Because travelers make important choices that affect the distribution of flows
(such as whether or not to travel, and where to go),
all assignment schemes
should be based, explicitly or otherwise, on a model as a behavioral principle;
when it is translated into a statement about flows it is an assignment
principle.
The major theoretical advance described in this report is a new assignment principle.
This principle combines the travel budget theory of Zahavi
(1979) and others (Kirby, 1981) with a detailed representation of a network.
It extends the current state of the art by combining demand, mode split, and
route assignment in an integrated formulation.
Because this formulation is
based on the travel budget theory, it has solid empirical backing.
In Section 2.1, we review the user-optimization assignment principle of
Wardrop (1952).
We show the relationship between an assignment about the be-
havior of individuals in a transportation network and the widely used statement about flows in the networks on which most modern assignment computation
techniques are based.
We state a new behavioral principle which is not new, it is the basis of
Zahavi's (1979) UMOT (Unified Mechanism of Travel) model.
from its new context.
Its novelty comes
The new principle is based on the concept of travel budgets:
travelers and potential travelers are assumed to have certain maximum daily
expenditures of time and money for transportation.
These budgets depend
on such attributes as household sites and household income,
Following the
same development as for user optimization, a set of statements about flows
in a network is derived from the individual behavioral principle.
3
Two versions of the new assignment principle are presented.
The first,
described in Section 2.2, assumes that budgets are equal for all members of
each class of travelers.
The second, in Section 2.3, allows budgets to differ
The second formulation, which is more
for various members of the same class.
nearly in agreement with reality, allows the same traveler to choose different
travel patterns on different days.
Important features of the new principle
are discussed in Section 2.4.
The new principle differs from formulations based on Wardrop's principle
in many ways.
First, daily travel behavior is considered, not single trips.
That daily travel is important is indicated by the work of Clarke et al. (1981)
whose empirical evidence shows that trip taking behavior must be treated in the
context of household size and age, and of the full day's transportation needs.
Consequently, instead of assuming fixed origins and destinations, we assume a
fixed origin, the traveler's residence location.
Travelers are assumed to have
daily journeys that start and end at that point.
Second, following Zahavi,
travel is viewed not as a disutility, but rather as a utility, providing
travelers with access to opportunities, and paid for out of limited budgets.
2.1
Review of Wardrop's Assignment Principle
Assumptions on Individual Behavior
2.1.1
Wardrop (1952) asserted two principles which were intended to characterize the bahavior of travelers in a transportation network.
One, which has
since been called user optimization, is based on the premises that
a.
Travelers have fixed origins and destinations.
b.
Travelers seek paths which minimize their travel time.
This principle has been of great importance in the development of
transportation models.
Its evident limitations have
provoked many researchers
to suggest extensions to include elastic demand (Florian and Nguyen, 1974)
multiple modes (Leblanc and Abdulaal, 1980) and other modifications.
4
2.1.2
Resultant Statements about Flows
To calculate flows, the above statements about individuals must be
translated into a set of statements about flows.
For the purposes here,
it is convenient to discuss path flows; these statements can also be expressed
in terms of link flows (Dafermos, 1980).
Let i be an index representing an origin node and a destination node
Let D i be the demand, in vehicles per time unit (most
in the network.
often hours) for origin-destination pair i. That is, D
wish to go from
a. (the origin) to b.
travelers per hour
(the destination).
Let j be a path
in the network that connects a. and b., and let J. be the set of all such
paths.
Let x.
be the flow rate of vehicles in path j.
The flow rates satisfy
x. > 0 ,
7
j e Ji'
x. = D
all j,
(2.1)
i
.
(2.2)
jeJ.
1
Let t. be the travel time of path j.
This travel time is the sum of
the travel times of the links that constitute path j.
The choice of the
shortest path can be represented by considering the flows and travel times
of any pair of paths.
Let j, k e J..
Then,
xj > 0,
xk > 0
=>
tj > t
=> x=
0.
j
k
D
tj .
(2.3a)
(2.3b)
That is, since travelers choose the shortest path, if both paths have
flow, they must have equal travel time.
If one path has greater travel time
than the other, no travelers will take it.
Path travel time is the sum of link travel times:
t.
=
EAjQ-
(2.4)
where TQ is the travel time of link Z,and AjQ is the path-link incidence
matrix x.
That is,
I1
if link Z is in path j,
g
jz~~~~~~~=
~(2.5)
~~~A
otherwise .
I0
Link travel time is a function of link flow.
vehicles on link Z.
f
=
Let fz be the flow rate of
Then,
Ajxj
(2.6)
.
This is, the flow on a link is the sum of the flows on all paths that pass
through that link.
Finally,
2.1.3
QZ is a function of f, the vector of link flows in the network.
Limitations
features of traveler
This principle neglects certain important
behavior.
In reality, origins and destinations are not fixed; the decision
if and where to travel (i.e., the demand) depends on congestion which
Demand also depends on the
in turn depends on the demand on the network.
money cost of travel which is not considered at all.
Ways of treating some of these features have been proposed by many
However the resulting formulations often require parameters
researchers.
that are difficult or impossible to obtain.
They also require a new calibration
for each application, so they have limited predictive value (Zahavi, 1981).
2.2
New Principle - Deterministic Version
2.2.1
Assumptions on Individual Behavior
Many authors have observed certain regularities about travelers' behavior.
(Kirby, 1981).
In developed countries
throughout the world, the average
traveler spends between 1.0 and 1.5 hours every day in the transportation system.
Furthermore, travelers who own cars spend about 11 percent of their income on travel,
and those who do not own cars spend 3 to 5 percent (Zahavi, 1979).
6~
~ ~ ~~ ~~~~~~~-·-'·
This must
· ---· ---·'· ----- ;-· ·-- ·------ ·-·· ------··-
influence the amount and kind of travel that each individual uses.
A behavioral principle that takes these empirical observations into account
must treat daily travel, not single trips.
This is because the travel time
budget is an amount of time spent each day.
We define a journey to be the path
in the network that an individual takes during a day.
It is made up of several
trips.
In the course of day, most travelers start and end their journeys
at the same point:
their residences.
(A small number of people do not,
including travelers starting or returning from business or vacation trips.)
These journeys need not be simple loops
for lunch;
For example, some workers go home
.
some people travel to work and home, and then out again for shop-
ping or entertainment.
We assume that such journeys are feasible for an individual only if the
total travel time is less than his daily budget T, and the total money cost
is less than his budget M.
The money costs include fixed costs for a car as
well as running costs if he drives; or fares if he takes transit or, of course,
both if he takes a car for part of his journey (evening shopping or entertainment) and transit for the rest (work trips).
Here, we assume that each traveler has fixed budgets which do not vary
from day to day, and which are the same as those of all others in this class.
In Section 2.3, we relax these assumptions.
Classes are defined as sets of travelers with the same origin node
(residence location) and with the same budgets for time and money to be
spent each day for travel.
These quantities
are determined by such socio-
economic indicators as household income and number of travelers per household.
(Zahavi (1979) asserts that one's daily travel time and money budgets
are influenced by such factors as travel speed; we treat both budgets as
fixed at this stage of the analysis.)
Of all the journeys that satisfy these budget contraints, which will
be chosen by a traveler?
their access to spatial
Zahavi suggests .that travelers attempt to maximize
and economic opportunities.
He indicates that
the daily travel distance may not be their precise objective, but is quite
adequate as a first approximation for a given urban structure and transport
network.
7
Because we are treating a detailed representation of a network, we can
represent other objectives.
For example, we can assign a value to each link
and each node in the network, and add the values encountered on a -journey
to yield the value of the journey.
These values can differ from class to
class: that is, the value of a given node (e.g., the location of an extremely expensive shopping mall) can depend on whether the traveler is rich or
poor.
Other formulas for calculating the value of a trip are also amenable to
this analysis.
To
summarize:
a traveler
in class a chooses a journey p, among all
journeys pa available to him, to
maximize
pep
wp , the value or utility of journey p
(2.7)
a
subject to the time and money budget constraints:
t
< Ta
(2.8)
m
< Ma
(2.9)
p -
2.2.2
P-
Resulting Statement about Flows
Equations (2.7) to (2.9)
by themselves are not adequate to characterize
network flows. For this purpose,they must be expressed in a form which is
analogous to that of Section 2.1.1:
a set of equations, inequalities, and
logical relations involving flows.
Let x
be the flow of class
p
certainly satisfy
x
a travelers
on journey p.
Then, x
P
must
> 0,
(2.10)
P -
Z
x = Da ,
pepa p
where D
below, D
(2.11)
is the total number of travelers available in class a.
can include people who will choose not to travel.
As we indicate
and Da
We measure x
p
in units of travelers per day.
To characterize flows, we must specify a set of relations that are consistent with (2.7) to
(2.9).
That is, if the demand Da is changed by a small
amount, representing one more traveler, the new distribution must-be consistent
with the behavior of the new traveler.
Assume that for each class a, the journeys p E P
increasing utility.
That is, if P
> P2 , w
2
'
.
>w
p
have the same utility value and that D a > 0.
a
are indexed in order of
We assume that no two journeys
P2
(Pairs of journeys are allowed to have
the same utilities by Gershwin, Orlicki, and Platzman, 1981.)
The journeys p e pa are divided into four sets:
1).
Infeasible journeys: are those for which
> Ta
t
p
and/or
> M
m
P
If p is an infeasible journey, x
2). Constrained
t
= Ta
P
= 0.
journeys are those for which
and
m
< Ma
m
= Ma
p-
·or
t
< Ta
and
P--
3).
p
There is at most one special journey p* such that
xp* > 0,
t p* < Taa
p
If no special journey exists, then we define p* as the smallest index of the
constrained journeys.
4).
Unutilized journeys are those for which
p < p*
(i.e., w
< wp*)
and
x
p
=
0.
9
To demonstrate that these conditions are consistent with (2.7) -to(2.9),
add a small increment 6D
to the demand D a .
Then,the time and money costs
associated with whatever journeys P are chosen by the new users comprising
ODa are t
+ at ,and mp + Sm
respectively.
Consider the effect of St
and am
P
Assume that 6tp > 0,and
mP > 0.
on the four possible sets of
P
journeys:
1).
and/or t
P
2).
Infeasible journeys remain infeasible since m
P
+ dt
> t
> Ta.
P
P
P
3
x
P
> m
p
> Ma
p
= 0.
P
Constrained journeys may not accept more flow and still remain
feasible since t
>T a
+ at
p
3).
Therefore,
+ &m
P
5
Therefore,
.
x
= 0.
p
P
If a special journey exists' it accepts more flow since
t* + at < T
p
pm* + Sm < Ma
P
P-
for sufficiently small 6D
.
Note that
D a may cause the special path to
become a constrained path.
Le
If there does not exist a special journey for the original equilibrated
system, new flow is then assigned to an unutilized journey which then becomes
a special journey.
4).
Unutilized journeys remain unaffected since the newly reduced flow
will be assigned to the higher utility, still available, special path.
2.3
New Principle - Stochastic Version
While the principle in Section 2.2 captures a greater variety of phenomena
than the user optimization principle of Section 2.1, there are several important
10
features with which it cannot deal.
The first difficulty is the fact the different members of the same socioeconomic class may have different budgets on the same day, and that the same
person may have different budgets on different days.
Zahavi (1979) shows that
there is a consistent coefficient of variation in both budgets among a wide
variety of populations.
Therefore, we redefine a class to be a set of people with the same residence
location and with a common probability density function for time and money budgets.
The probability that an individual of class a will have a time budget between
u and u + Su and a money budget between n and n + Sn is
fa (u,n)6u Sn
(2.12)
The number of travelers of class a who have time budgets between u
and u + Su and money budgets between n and n + (Sn is
Dafa(u,n)Su
n .
Let R be a region in (u,n) space.
(2.13)
The number of travelers whose budgets
fall in that region is
Da
fa(u,n)u Sn.
(2.14)
R
To state an assignment principle, we relate flows to integrals of the
form (2.14).
pa
Let
a
= {P'
a
a
P2,'''
Pk
be the set of journeys available to travelers
in class a and, following the
convention stated in Section 2.2, let pl be the least desirable journey and
Pk be the most desirable. The utilities of all the journeys are assumed to
a
a
be distinct. Let xl,... ,xk be the flows on those journeys and define
k
a
X.
=
xa
.
(2.15)
a
That is, Xa
journeys p
X. is the total demand on journeys
a
a
and m be the time and money costs of journey pi'
a
a
pai+ll
~Let
a
Pki
ta
Leta a
i
Since pk is the most desirable journey, the total flow on-this journey
is the number of travelers whose
time budgets are greater than,or equal to,
tk,and whose money budgets are greater than,or equal to,m
Xk = Da
That is,
f(u,n)du dn .
(2.16)
na
n_>m k Define
Ra
fnu~n~lu > a
Rk = {(u,n)u >
Uk
}
n > m k},
(2.17)
= Rk
(2.18)
This region is illustrated in Figure 2.1.
Using the new notation, (2.16) can
be written
Xk = Da
f(u,n)du dn .
(2.19)
Uk
To characterize the rest of the flows, we define
a
R-
a
{(u,n)lu > t?,
a
n >
(2.20)
(2.20)
the set of expenditure levels that equal or exceed the required expenditures
on path i.
The people who can afford journey Pi or better are those whose
budgets fall in any region R., i < j < k.
a
Ua =
R
Define
a
(2.21)
.
j>i
Then the number of travelers X. that take journey p. or better is the number
1
1
whose budgets fall in region U a .
Xa = Da
That is,
f(u,n)du dn.
(2.22)
U1
Equations (2.20) to (2.22) are a complete statement of the assignment
principle in the stochastic
case.
The statement is remarkably concise
12
m
(tk, m k )
t
Figure 2.1:
Region of Integration for Most Desirable Journey
13
compared with that required in the deterministic case, especially in light of its
greater information content.
In addition, a simple numerical technique for
solving (2.20) to (2.22) suffices (Section 3).
Equation (2.21) can also be written
U
a
=
1
a
a(2.23)
i '
0 < i < k-l
Ui+l
U
i+l UR.1
Figure 2.2 illustrates Ua 1
i+lg
Ra
i
(2.23)
and U a.
sat
Note that the boundary of U. always
i
has a staircase-like structure, with a vertical half-line at the left and a
horizontal half-line at the right.
The flow on path i satisfies
a
a
i
1
xa
1i+1'
so that
i
D
Ua
1
i
a
(2.24)
fa(u,n)du dn .
i+l
a
a
is the oddly shaped rectangular polygon in Figure 2.2
The region U. - U
1
i+l
whose lower left-hand corner is (ta , ma).
a
a
a
Note that if t. and m. are sufficiently large, R ifalls entirely inside
a
a
a
a
This is intuitively
- U i+l = 0 and X = 0.
In that case,U
of Uil
reasonable.
Journey i is the least desirable of those currently under considera-
tion (i, i + 1,...,k) because of the ranking contention.
If both its travel
time and its money cost are greater than those of all better paths, it attracts
no flow.
Equations (2.20) to (2.22) have been constructed
to
be consistent with
The region Ua contains all budgets that are greater than,or
a
a
a
'Pk · Equation (2.22) implies
equal to,the expenditures on journeys p, Pi+l, ..
(2.7) to (2.9).
that if an individual has budgets that are in U., he will choose one of these
a
a
a
Pi 1 .'
journeys and not journeys pl, P2 ,''-
14
ui+1
L
boundary
m
u, 0Q
li
Figure 2.2:
· t~~~~~l~
Region of Integration for Ordinary Journey
15
In particular (2.24) implies that if his budgets fall in Ui- U
a Ua i+l'
he
a
will
pa.
I f he
afford
to take what
journey
but he cannot
Pa+1 take
* Pp journey
he will
choose
pancan
This
is e::actly
the Pi'
formulation
(2.7) afford
to
Pi+l' ''''Pk h e'
wi' choose
Pi'
a
2.
(2.9) says:- pa is the best journey (the ordering convention requires that the
utility values of pa,' ' ,pa-1 are less than that of p.) whose costs are less
than ,or equal to,his budgets.
2.4
Discussion of New Principles
There are several features of the new principles that are not determined
by the reasoning presented above.
all the examples to follow.
Certain choices have been made in constructing
These choices are described here.
Alternative
choices could easily have been made with no difficulty.
2.4.1
Null Journey
It is convenient to introduce a null journey for each class.
This is a
journey that requires no travel time or money, that has less utility than any
other journey available to its class, and whose flow does not affect the
travel or money costs on any other journey in the network.
The people who
take the null journey are the people who stay home; their budgets are insufficient to allow them to do any traveling at all.
2.4.2
Costs
In the numerical examples to follow, the travel time for a journey is
the sum of the travel times on its constituent links.
The money cost of
travel is computed in the same way with an additional term due to either the
fixed cost for owning a car or a fare for transit.
Neither the statement of
the equilibrium conditions nor the algorithm in Section 3 depends on the method
of calculating the costs.
2.4.3
Utility of Journeys
We have assumed that utility is constant, depending on the geography
of the network, and not depending on flows, delays, or money costs.
We assume
that the utilities of journeys available to the same class are distinct, and
that journeys may be ranked in order of utility, with the null journey
worst.
16
the
In this model, what is important about a journey is not its utility value,
but rather its utility ranking.
It is important that one journey be better than
another; not how much better.
This feature obviates a precise determination of utility.
mination may be expensive or impossible.
least on an individual basis:
Such a deter-
It is a reasonable assumption at
each individual chooses the best journey he can
afford, without being concerned with how much better it is than other journeys.
On the other hand, different individuals may rank journeys differently, particularly if they are similar.
In the examples discussed below, it is assumed that utility, like travel
time, is accumulated as one traverses a journey.
A utility value is assigned
to each node, and the value of the journey is the sum of the utilities of the
nodes passed through.
This choice
ideas.
is made in an effort to build on, and further refine, Zahavi's
Zahavi (1979) suggests that a reasonable approximation to Jtravelers'
behavior is to assume that they maximize the distance they cover within their
time and money budgets.
He points out that distance is only a surrogate for
travelers' real objectives, which are to work, shop, be entertained and generally
take advantage of as many of the facilities in the region as possible, within
bounds of time and money constraints.
The utility value of a node is simply
a measure of the number of these facilities that can be found at each location.
We say that utilities are additive if the utility of a journey is the
sum of the utility values of the nodes and links through which the journey
passes.
Again, neither the statement of the equilibrium conditions nor the algorithms
presented in the next section depend on the formula for calculating the utility
ranking.
17
2.5
Related Work
There are similarities between the model suggested here and a model
discussed by Schneider
(1968) and Dial
(1979).
In their model, both time and
money costs are important, but individuals do not have budgets.
traveler seeks to choose a trip
Rather, each
(not a day's journey) i that minimizes a dis-
utility which is a linear combination k tti+k mm
i..
Travelers differ by having different coefficients k t and ki. This principle
a .
leads them to a graph like Figure 2.2, but where U. is replaced by a convex
polyhedron.
Trips that have both greater time and greater money costs than
other trips get no flow.
This is also true in the present model unless such
journeys have superior utility values.
3.
3.1
SOLUTION TECHNIQUE FOR STOCHASTIC VERSION
OF NEW ASSIGNMENT PRINCIPLE
Statement of Equilibration Procedure
Let x be the vector of all path flows in the network.
Define g(x) to
be the vector of the same dimensionality as x whose component corresponding
to journey i of class a is
gi(x)
Da
fa (u,n)du dn
i
The dependence
(3.1)
i+l
of g on x is
due to the dependence of Ua and Ua
on x.
These sets depend on x because they are determined by the time and money costs
a
a
of journey Pi and of all the journeys better than p
for class a. These costs,
in
turn, depend on the costs of the links
link costs depend on the link
flows that
on all
Equations
(2.20),
in
make up those journeys,
flows, which are the sums of all
pass through those links.
the journey flows
that
It
is
clear that, in
and the
the journey
general,
ga depends
the network.
(2.21), and
(2.24)
(which together are equivalent
to (2.20) to(2.22)) can be written
x = g(x).
(3.2)
Thus,we seek a solution to a fixed point problem.
It is observed that
g(x) is a continuous function on the compact set
xa > 0,
1x
i
a
(3.3)
a
= Da
(3.4)
1
to which x is restricted.
exists
Consequently,
at least one solution to
(3.2)
(Dunford and Schwartz, 1957),
Equation (3.2)
suggests a procedure;
x(O) specified,
x(q+l)
=
(3.6)
(l-X)x(q) + Xg(x(q)),
19
where
O < X < 1 .
In our experience,
(3.7)
(3.5),
(3.6) converges whenever X is sufficiently small.
Convergence is considered to have occurred when
max
a,i
x.a(q) - gi(x(q))l < e
(3.8)
The main effort in executing (3.6) is evaluating the regions U., and then
performing the required integrals.
3.2
This is described in the following section.
Calculation of Integrals
It is.convenient to write the iteration process as
e.
(q+l)
= (1-X)a(q)
+
fa(u,n)du dn,
X
(3.9)
U. (q)
a
a
a
x. (q+l) - X(q+l) - X
(q+l),
i+l'
1
.
(3.11)
where X.a(q) is the class a flow on journey pa or better.
The index k refers
I
to the best (highest utility) journey for class a and, as usual, the journey
index numbers increase with increased utility.
In Section 3.2.1 ,we characterize the regions U.a (q).
1
In 3.2.2,we demon-
strate how to evaluate the integral in equation (3.9).
3.2.1
Regions of Integration
Region U. is defined by equations (2.18),
the argument q is suppressed).
(2.20), and (2.23) (where
It always has the characteristic staircase
shape of Figure 2.2, of which Figure 2.1 is a special case.
the integration in (3.9), a list of the corners of
required.
The corners of U
i+l
are displayed in
To perform
Figure 2.2 is
Figure 3.1.
20
-----------
,;
_,_............
... .,...,_.._...,
_~~._~~~~~~~~~_
~
~
·---- · -----
,-----
·---
·- -·---------------------·-·------..
rI -(Uu
2 , nl)
(u 3 , n 2)
(u 4 , n 3)
(u4 X
l,
/--(U5,
n 4)
X
(Ut,'_'
! ,
/_ _,:-(u6 ,n 5)
L /__
(t ,. mP.)~-f-
-
...
-
(U 4 ,n 4 ) U5 , n 5 )
(U 6, n 6 )
_igure
Corners
3.1:of Ui~l and
Figure 3.1:
Corners of U +
21
and U.
t
The components of a corner of the form (u.,
n.)
are the travel time
and travel money costs of some journey of class a better than journey i.
Heretofore,they have been listed in order of increasing utility.
To perform
the integration, they must be listed as they are in Figure 3.1, in order of
increasing travel time and decreasing money
(or vice versa) .
the costs of journeys i such that R a is a subset of U
~1
in this list.
~
~i+lmust
Furthermore,
not appear
(See the remarks following (2.24).)
Let La be the list required to perform the integration over U.
1
1
It is
defined inductively from La
based on (2.23). From (2.17) and (2.18), we define
i+1
a
[ta
a)]
(3.12)
where Pk is the best path.
a
[(uln
Let
(u
,, ,n
)
)
(3.13)
where
u1 < u2 <
' < ur,
n1 > n2 >
> n
1
2
(3.14)
r
and
(3.15)
.
If, for some j, 1 < j < r
t
a
1-
> U
]
(3.16)
a
m. > n.
1-
j
22
then,
L
= La
i+1
i
(3.17)
This is because journey i has lower utility than the journey whose costs
are (uj, nj).
Since its costs are also greater, there is no reason to take
this journey.
Geometrically, R a is a subset of Ua
,and (tfalls
1
i+l
z
1
inside
of U a
z+l'
Otherwise, let a be the largest integer such that
u
(3.18)
< ta
8
and let
n
be the smallest index such that
(3.19)
< ma
(In Figure 3.1, a = 2, 8 = 4.) Then,the list L-. is constructed from list
a
a
a
with (ti,+i).
n+),...,(u, n)
Li+l by replacing all the points (ul,
That is, if
La
i+l
i+l ==
i+l
Cl
1l(c I...Icr
ci
1 ,
L_ =
. ..
,
i+l
r :i
c
i
(3.20)
]
then,
i
]
i+l
3
c
= Ct,
(
Ca+=(l
m
i
i+l
cj = c--2+
j
s
'
j = a+2,...,s
r-f+ai+2
In Figure 3.1,
i+l=
Lia
=
[(u1 ,nl)'
[(uln
1)
'
''
(U6 n'6 ) ]
(u2 ,n2 )
(u,m
(t
5 ),
'
ma
(u4
(u6 ,m6 )]
23
(3.21)
Note that the numbers of corners in U. (i.e., s) may be greater than the
number in Ua+ 1 (i.e., r) by 1, or it may be equal, or it may be fewer if
i+l
6 > a+2.
3.2.2
Calculation of Integrals
Once list La is determined, the integral in (3.9) can be written
s-1
fa(u,n)du dn
=
a(u,n)du
f
dn
U
Si
+
where s is the number of corners of U.,
Sj = {(u,n)u
(3.22)
a(u,n)du dn,
< u < u+
1,
n >
S. is the strip.
}
(3.23)
and Q is the quadrant
Q = {(u,n)lu > us, n > n}
(3.24)
In all the examples described below, we have assumed that the budgets
are independent.
fa(u,n) =
As a result,
(a(u)4a(n),
(3.25)
and
(u,n)du dn =
jfa
f
(u,n)du dn
=
[
L
a(u)d d
a (u)du
The integrals in (3.26),
P and
1
[
a(n)d
(n)dn
(3.26)
(3.27)
(3.27) are particularly easy to calculate when
density functions are piecewise linear.
24
the
We have observed that numerical
results are not sensitive to the detailed shapes of these distributions
although they are sensitive to their means and variances.
3.3
Algorithm Behavior
In the following sections, we discuss a number of numerical examples.
examples are presented in Gershwin, Orlicki, and Zahavi, 1981.)
(Other
We restrict our
attention to the behavior of the solution of the system (2.20) to (2.22).
Here,
we present an informal summary of our observations on the behavior of the
iteration procedure (3.6).
The convergence properties of the algorithm are sensitive to two sets of
quantities:
A, and the variances of
Xa
( ) and ka( ).
The larger the variances, the greater the reliability of convergence.
When the variances are small, it is very easy for the algorithm to overshoot
the equilibrium distribution and oscillate wildly.
We conjecture that this behavior is due to the fact that, when the
variance is small, the travelers in a given class are similar to one another.
If the cost of a given journey is too high, all the members of
that class will
react in the same way, and most of the flow will be removed from that journey
by the integral in
(3.9).
If
a journey, the integral in
the costs are too low for the present flow on
(3.9) willtend to redistribute most of the flow of
that class onto that journey.
If the variances are large, only a small amount
of flow will be affected by an error in cost.
The step size X also affects algorithm behavior.
When X is
small,
the change from iteration to iteration is small, and although convergence is likely
it is time-consuming.
When x is large, overshoot is a danger, and oscillations
can be observed.
The amount of computer time that the algorithm requires is also greatly affected by the computation required to evaluate the integrals on the right-hand
side of (3.26)
and (3.27).
Distributions that require a great deal of airthmetic
(such as a normal, Erlang, or gamma) require much more total computer time
than others (such as the uniform or a piecewise
25
linear density).
It is clear that these observations can be combined to enhance the
efficiency of the algorithm.
First, X can be increased as the algorithm
shows signs of slowing down; i.e., reaching equilibrium.
Second, the variances
can be initialized at much larger values than the required variances, and
decreased gradually when the algorithm seems to be converging.
Finally, if
it is necessary that results be computed with difficult-to-calculate distributions, the algorithm can be restarted after first converging with an easy
distribution.
26
4.
NUMERICAL EXAMPLES
In this section, we describe several examples of small-size networks
that we analyze using the numerical technique of Section 3.
These examples illus-
trate various qualitative properties of the assignment principles of Section 2 and
of the iteration process,
A network of more realistic size and complexity is
presented in Section 5.
4.1
Four-Link Network
There are five journeys in the network of Figure 4.1:
the four that each traverse one link once.
traverses link i, is xi.
the null journey and
The flow that takes journey i, which
The flow on the null journey is x0 .
The four links
have identical delay functions given by
t.
+(-L
=
4 .X.
(4.1)
+25
ti
The journeys have different utility values:
journey 4 is better than journey 3,
and so forth.
Figure 4.2 displays the flow levels as a function of total demand when the
time budget has a triangle distribution with mean 2.00 and variance 1.95.
triangle density function is displayed in Figure 4.3.
The
The money budget is set
high so as to be a non-binding constraint.
When the demand is near zero, nearly all flow is attracted to journey 4.
Journey 4 is the best journey and the delay is small, so nearly all travelers can
afford to choose it.
As the demand increases,the flow on journey 4 increases.
However, some
travelers are excluded from it since its delay has increased, so they take
journey 3.
Eventually,journey 3 becomes expensive and travelers move to journey
2, journey 1, and the null
journey.
In the limit as D-*o,
Xi + c*,
1
(4.2)
O + D - 4c*,
(4.3)
27
2
Figure 4.1:
Four-Link Network
28
0
0-z
w..L
0
0
rr,
o
00
00
cu~~~~(
0
0
0
o
>:
o
o
0i
0
o
!
b,
EN
I
o0o
0
(0
0
I0
cZ
o
rOc
r~o
0o
~
0
0
0,'
,~'~
,1 c
~
V.
0
0
N
0
0
0
(0
N
'
0
01
0(0
'N
N
N
N
-
(AVO/S3-1':AVUJ.) MO-I-!
29
rZ4
Ficure 4.3:
Triangle Density Function
30
------ 1-- -- ·
where c* is a limiting flow, and x 0 contains all the travel demand that is
unmet.
Figure
4.2 illustrates the relationship between the deterministic and
stochastic versions of the principle.
When the demand D is as indicated by
the arrow, two journeys are approximately at budget levels (constrained
journeys), one has less travel time than the budget level (special journey),
and two have nearly no flow(unutilized journeys).
In the deterministic case, Figure 4.4 describes the behavior of the flows.
Again, limits (4.2),
T = .
(4.3) are valid.
+
Here c* is the solution to
4,
(4.4)
where T = 2.00 is the value of the time budget so that c* = 293.5.
Clearly,
at the arrow, journeys 4 and 3 are constrained, journey 2 is special, and
journeys 1 and 0 are unutilized.
4.2
Small Network with Interacting Journeys
Figure 4.5
is a network with three nodes and five links.
The link time cost functions are given by
numbers are indicated.
TI ( f) = Tzl
+
Node and link
2
(4.5)
and the link money cost functions by
V%(f)
=
V1
+
.5
(4.6)
where the parameters are listed in Table 4.1.
Several examples
are treated that are
the budget density functions 0 and G are
based on this network.
uniform.
31
In each,
0
0
o
00
0
00
0
000O
Oo
00
0
N0
~32~
©
o
o
0
.~
"
0
Figure 4.5:
Three-Node Network
33
Table 4.1 - Network Parameters
link Z
lR2
Z
1i
2oO
0
1
0.25
2
O.25
.
o,3o
400
Q.SO
4
0.50
jO0
.0
5
0.50
200
0
.o
Case 1
The time budget
is uniform between 2.0 'and 2.5 and the money budget
uniform between 3.0 and 3.5.
The total demand
is 200.
is
Table 4.2 lists the
journeys in increasing order of utility, as well as the equilibrium flow,
travel time, and money cost.
Table 4.2 - Results of Case 1-
Flow
Journey
Time
Cost
null (1)
0
0
0
: - 3 - 1 (2)
0
1.03
.74
1 - 2 - 1 (3)
114.31
16d
1e61
85.69
2.29
2.64
I - 2 - 3 - 1 (4)
Journey 4, which is the most desirable, is like a constrained journey in
that there are some travelers on it who are spending their entire travel-time
budgets.
Journey 3 is analogous to the special path of the deterministic
formulation since no traveler is spending either of his full budgets on travel.
Journeys i and 2 are unutilized.
It is noted that this interpretation of the
relationship between the stochastic and deterministic principles does not
always hold.
It works here because the uniform density is zero outside of
a certain range.
Case 2
Case 2 is the same as Case 1 except that the demand is raised to 300.
Results are presented in Table 4.3.
34
Table 4.3
Journey
Results of Case 2
Flow
null (1)
Time
0
Cost
0
0
1 - 3 - 1 (2)
97.12
1.20
.93
1 - 2 - 1 (3)
176.24
2.16
2.29
26.65
2.46
2.87
1 - 2 - 3 - 1 (4)
In the previous network, when the demand is increased, it fills up the current "special" path, and then overflows onto the next. Here, all the flow
is redistributed. In particular, there is less flow on the best path in Case 2
than in .Case 1.
The reason for this is that the time required to traverse journey 4 has
increased from 2.29 to 2.46, which is almost at the upper limit of the timebudget distribution. This increase in time is due to the increased flow on
journey 3, which shares link 1 with journey 4: and on journey 2, which shares
link 5. Again, the stochastic version of the assignment principle mimics the
deterministic in that there are constrained, special, and unutilized journeys
in the proper order.
Note that the flow on journey 4 is now smaller than it
was in case 1.
Case 3
It is observed that journey 2 has time and money costs which are less than
one-half of the upper limits on the budget distributions.
fore be expected to go around the loop twice.
additional journey:
1 - 3 - 1 - 3 - 1.
Some travelers might there-
Consequently we have added an
The utility of this journey is greater
than that of journey 2 and less than the utility of journey 3.
To our great surprise, no redistribution takes place after the new journey is
added.
That is, the new equilibrium is such that the flow on the new journey is 0,
and the flows on the othrs are the same as those of Table 4.3.
The time cost of
the journey, 2.40, is greater than that of journey 3. (2.16) although its utility
is less.
(The money costs are irrelevant since they are less than the lower limit
of the money budget density function).
Consequently, no travelers who are capable
35
of switching from 1 -2
- 1 to 1 - 3 - 1 - 3 - 1 (i.e.,
those on 1 - 2 - 1 wnose
time budgets are greater than 2.40) have any incentive for doing so.
This indicates the marketing aspect of providing travel service.
new travel alternative is offered
(in this case, a new journey),
When a
it is not
enough to ascertain that it is better than some existing alternatives that are
used, and that it is affordable by
(i.e., within the budget of) some travelers.
The new alternative will be used only when there are some travelers that can
afford it, and for whom it is better than their current transportation choice.
This result may be seen in terms of the integration regions U..
4.6
Figure
shows the uniform joint-density distribution of budgets for travelers
in the network and the Ui regions for journeys 2, 3, 4, and the new journey.
The integrals of regions of the budget joint density are performed in order'
of utility, with the region for the highest utility journey evaluated first.
The new journey we propose has utility greater than journey 2, and less than
those of journeys 3 and 4 of the original network.
Figure 4.11 shows that by
the time the integral of f(.)
over the region U
for the proposed flow is
new
evaluated, all potential users of the new journey have already been assigned
to journeys 3 and 4.
The travelers having budgets in the joint density con-
tained in the region U 3 - U2 are the only candidates still unassigned, and they
cannot afford the new journey.
Thus, even though a journey with higher utility
than a presently employed path is available which has costs in the feasible
budget region, it may go unused depending on its value to travelers relative
to other journey choices.
4.3
Circular Network
Figure 4.7 contains a network consisting of 13 nodes and 40 links in
the shape of two concentric circles.
in Gershwin,
Orlicki, and Zahavi
The details of this network are presented
(1981).
Here, the effect of the shape of the distribution is investigated.
Two
runs are compared that are identical -- 5 journeys for each of two classes; time
budgets means are 1.25 hours and variances 0.5
are 3.00 dollars and variances 1.5
(hour) ; money budgets means
(dollars) --in all respects except one.
run has gamma distributions and the other has uniform distributions.
36
--·-·---·---- ---- ---·---·-·--·-·-·-----·-··
F._._,~,... _...._.r
. ---U---~~~;-1
One
See Table 4.7.
f(t,m) >O
(t ,m
l)
U3
U
(t 3 7m 3 )
2 -
(tnew, mnew)
U2
(t2am
i,
,
!i,
1
Figure 4.6::.
a)
,I
!
2
i
3
Regions of Integration for Cases 2 and 3
37
i,,
4
Jg
t
5
Figure 4. 7:
Circular Network
38
E-4
UO
(NO
y)
(I,_
O
0
O
!
Q ri
\
00
I(o
iNf N LA
(N
Oq
0
CO
O0
oN
_4
r
*,
7
0
_H I0
I
r-
OI
e'l
O(N
N,l
I~
C'
I
a'
15
r
'-40
I
~.I4
II
N
L
r
0
V
1-
I
a
I
~~rO ri
~
3
The resulting corresponding journey costs are nearly identical.
corresponding flows are also close to one another.
The
If larger variances were
used, we assume that the flows would be even closer.
(Larger, more realistic,
variances could not be used because one of the distributions was uniform.
There is a limit on the variance of a uniform distribution whose mean is specified
and whose lower limit is required to be zero or positive.)
This assumption
follows from the observation that the journey costs are similar in the two
cases and from our experience that large variances tend to reduce the sensitivity of the distribution of flows to journey costs.
40
~ ~
'~~~~`~~~-'
~ ~ ~~~~~~~`
~ ~ ~~^"'I"~~"~;"`'
~ ~ ` ~~~~-~
I~-~~
·-"~~··-·--·-
·
··
1
-
5.
LARGE NETWORK
In this section, we show that a system of realistic size and complexity
can be treated using the technique described here and that it yields reasonable
results.
Relatively crude models for mode delays and costs are used.
parking fees are not included.
For example,
Only a single kind of car is represented, and
a single global average of 1.5 occupants per car is used for all economic
classes and all residential locations.
Walking, which is an important trans-
portation mode in the dense downtown area, is not considered.
For details, and more variations on the basic network, see Gershwin,
Orlicki, and Zahavi (1981).
5.1
Discussion of System
5.1.1
Network
The network is presented in Figure 5.1.
It has 144 nodes and 264 links.
It is symmetric about its vertical and horizontal axes.
343.2 kilometers.
Its total length is
Lengths of links are indicated in the figure.
Roads are
most dense near the city center and become less dense toward the periphery.
All streets are one-way,
We calculate the utility of a journey by counting the number of nodes that
the journey passes through.
This count is multiplied by 1.1 for car journeys
to represent the attractiveness of a car over that of a bus.
equally desirable.
All nodes are
Because of the greater density of nodes near the city
center, journeys going inward are more valuable than those going outward.
Journeys are paths in the network that start and end at the same residential node.
Between 76 and 81 journeys are made available to each class.
No mixed-mode journeys have been considered.
41
.....
,i,
,
LoI
ITI
~
k
. ,
Lo
to0N
i
',
NE~O
(0I:
ni
;5
l
~
0__20
N
~ t- '
4'
-
(D
t
-o . ._
(
t D-CI
o' (0
0
O
r~cnlI,
n
.
o42C
I
0
0
N
N
Car journeys are allowed to go anywhere in the network as long as pathcontinuity and link directions are respected.
Bus journeys are more restricted.
They have the same freedom as cars in the inner city; that is, among nodes 40,
45, 100, and 105.
certain links.
However, outside of that region they may travel only on
These are the links that connect the following nodes:
13-24;
37-48; 97-108; 121-132; 2, 14, 26,...,134; 4, 16, 18,...,136; 9, 21, 33,...,141;
11, 23, 35,...,143.
5.1.2
Population
The population consists of 300,000 people divided into 2 income groups
and 16 residential locations.
The wealthier group has a household income of
35,000 dollars per year and the poorer has a household income of 15,000 dollars
per year.
Households are assumed to contain 3 people, on the average; and years
have, on the average, 320 days of regular travel, so these incomes are 36.46
dollars per day per person and 15.63 dollars per person respectively.
The locations at which people live and their class designations are listed
below.
Classes 1 to 16 are the wealthier people; classes 17 to 32 are the poorer.
-The number of people in each class is 9375.
Class
Node
14
16
21
23
38
40
45
47
1,
2,
3,
4,
5,
6,
7,
8,
17
18
19
20
21
22
23
24
98
9,
25
100
105
107
122
124
129
10,
11,
12,
13,
14,
15,
26
27
28
29
30
31
131
16,
32
43
Because of the great size of this system, techniques have been sought to
reduce computation time.
function
(Figure 4.3)
A very effective technique is to use a triangle density
rather than a gamma or other distribution.
involving the triangle are simple and, as we
Calculations
argued in Section 4.3, probably
accurate when variances are large.
The time budget distribution for all classes is triangular with mean 1.1
hours and variance 0.43
of variation of 0.6.
(hours)
for each traveler, which yields a coefficient
This is consistent with Zahavi's
(1979) empirical
findings.
The money budget is also triangular, with means of 11 percent, or 6.00 and 2.55
dollars per traveler. The variances are 17.50 dollars
and 3.10 dollars 2
respectively.
5.1.3
Time Costs
The time for a journey is the sum of the times for its constituent links.
In the case of bus journeys, an additional delay of 0.2
hours per transfer is
added to represent the time spent waiting for a bus.
Link travel time for cars is given by
t,
=t
1
(1
+ 0.15 C,
(5.1)
where tI 1 , and cI are constants.
The free-flow time, TI1
The capacity, c., is 24000 vehicles per day.
is the length of each link
(as indicated in (Figure 5.1)
divided by the free-flow speed, which is assumed to be 50 km/h.
Link travel
time for buses is assumed to be twice that for cars.
The flow, fi'is given as
=afc
f
+c
fbQ
(5.2)
b
c
and f are the total car-traveler flow and bus-traveler flow through
2
Z
link Z. That is,
where f
E
f
b
-
cr
E
rfZAi
.
A
(5.3)
ba
=
44
where the second sum in
(5.3) is over all car journeys, and the second sum in
(5.4) is over all bus journeys.
The quantities ac and ab represent the impact
of a single traveler on the total car-equivalent flow.
We choose
a
c
= 1/1.5
since we assume a car occupancy of 1.5 travelers, and
a b = 0.125.
This is obtained by dividing the bus equivalency factor (3) by an average
bus occupancy of
5.1.4
24.
Money Costs
The cost of most bus journeys is a fare of 1.00 dollars. There are a small
number of journeys that involve multiple loops.
That is, some nodes, other
than the residence node, are visited more than once.
We assume that travelers
tak:ing such journeys are making stops that do not allow free transfers, so
they are charged one dollar per loop.
The cost of a car journey is given by
m. =.5
F
in which m
1
D
T
F
+mtj),
(m + m
(5.5)
is the fixed cost of owning a car and is assumed to be 5.00 dollars/day.
This value has been chosen to account for the cost of purchasing a car as well as
other fixed costs such as insurance.
The divisor
1.5 is the
occupancy and reflects the occupants sharing the car's costs.
are due to the empirical findings of Evans and Herman
average
car
The other terms
(1978) who found that
the fuel cost of a car is linear in the distance and the time it travels.
We have increased their coefficients to account for other running costs such
D
as repairs and maintenance. We assume m is 0.0672 dollars per kilometer times
t
the length of a journey, and that m is 2.322 dollars per hour.
45
5.2
Basic Network Results
Table 5.1 lists summary results for this network.
people take cars than buses;
travel.
Note that more wealthy
more poor people take buses.
More wealthy people
Note that car behavior is similar for all people who take cars, and
bus behavior is the same for all bus riders; the major difference in behavior
is due to the different fractions of each class who take each mode.
Note that more people travel in the inner city (i.e.,
more of those who
originate at nodes 40, 45, 100, and 105) and more stay at home in the suburbs.
Wealthy people gain more benefits, i.e., utility, from travel, than poor
people.
They spend more money, and more of them take cars.
more time traveling.
Poor people spend
Note that all times and speeds are door-to-door times
and speeds.
5.3
Fare Change
Table 5.2
by 50 percent.
income.
summarizes the behavior of the network when bus fares are raised
Bus riders are profoundly affected, particularly those of lower
Car commuters are almost unaffected.
It is important to see that, in
this network, when fares are raised, people do not switch from buses to cars
(except for a very small number of the poor).
Instead, they switch from travel-
ing in buses to not traveling at all.
It should be noted that a crucial modeling assumption may have had a
profound effect on this result.
Travelers are assumed to be independent, with
independent time and money budgets,
In reality, members of a household may
share the money available to the entire household.
If this is the case, increas-
ing bus fare may have little effect on bus travel in households where some members
travel by car.
Instead it may increase the expenditures of the bus traveling
members which reduces the funds available for the car travelers.
Thus, it may
decrease car travel.
46
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5.4
Tolls
An experiment has been run to assess the effect of adding tolls on certain
The intention is to discourage the use of cars
links entering the inner city.
in the inner city, so only cars are subject to the fee.
The links are 45-44,
100-101, 40-52, 105-93, and the amount charged is 1.00 dollar.
As Table 5.3
indicates;
the desired effect is achieved.
travel by car, and those who do drive tend to drive less.
Fewer people
The effect of tolls
is greater on the wealthy than on the poor since it is the wealthy that drive
more.
The average daily money expenditure of wealthy car drivers goes up, while
that of all others is nearly constant.
The automotive travel of the poorer
people who do use cars, however, is affected much more than that of the wealthier.
Sus ridership increases enough so that there is only a small decrease in total
travel o
5.5
Prohibitions
Table 5.4 displays the effects of forbidding cars from traveling on certain
4inks.
The links are the same as those studied in Section 5.4.
'In fact,
the
method used is to impose a 50 dollar toll on those links, which very effectively
limits their use.)
The effect of prohibiting travel is only a little greater than that of
imposing tolls on the same links.
close to those of Table 5.3.
Most of the numbers in Table 5.4 are quite
One significant difference is the average daily
expenditure of wealthy car riders.
When a 1.00 dollar toll was imposed, enough
of them were willing to pay it that the expenditure went up.
dollar toll was unreasonable and they refused.
However, a 50.00
Because they lost opportunities
to spend their money, their expenditure decreased.
It seems reasonable to conclude that as the toll increases from zero
base case in Section 5.2)
the expenditure of wealthy car riders will increase,
reach a maximum, and decrease.
get distributions.
(the
The maximum expenditure is limited by the bud-
Additional study is required to verify this conclusion
and to make it precise.
49
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5.6
Computer Costs
The results described here were calculated using the MIT Multics timesharing
system on a Honeywell 68/DPS computer,
Typical runs required 5-8 iterations
when executed from scratch, i.e. when the initial flows were set to 0.
of iterations depended on the value of
were used.
X .
The number
Values of X in the range of .5-.8
Variational runs, such as those in Sections 5.3-5.5 typically required
only 1-2 iterations,
A typical
Such runs were executed with
X
near .9.
run took 40 cpu seconds for initialization and post-processing
and 22 seconds per iteration.
Memory requirements are dominated by the number and size of journeys.
For
this example, approximately 110 kilobytes were required.
Considering the substantial size
small.
of the network, these costs are remarkably
Four-fold symmetry was exploited in exploring this example but we see no
obstacle to the study of larger unsymmetric networks or the implementation of
these results on a smaller computer.
52
-----------
-
6.
CONCLUSIONS AND FURTHER RESEARCH
The demand/distribution/assignment principles described in Section 2 and the
algorithm presented in Section 3 promise to be valuable tools for the understanding
and prediction of travel behavior in a city,
the usefulness of these tools.
Additional research can enhance
This research is of three types.:
modeling,
analytical, and empirical.
6.1
Modeling
6,1.1
Time-stratified Travel
As Stephenson and Teply (1981) show, changes in network conditions can lead
to changes in the time of day that people travel,
Clarke et al (1981) demonstrate
that travel decisions vary during the course of the day and depend on the scheduling
of events such as work, school, and shopping hours,
Consequently, the variation
of the distribution of traffic in a network as a function of hour of the day can
be important.
In principle, this can be treated using the approach introduced here.
now become paths in space-time and not merely in space.
Journeys
That is, to describe a
journey, one must not only list the links and nodes that the journey passes through,
but the time of day that each link and node is reached.
Two journeys that pass
through the same points in the network are now considered different if they reach
those points at different times.
Utilities may be assigned according to when a
visit to a node or link takes place.
(This is important:
food shopping after work
is worth much more than shopping before work if there is no intervening stop at
home).
Link costs vary with the time of day according to the time-varying conges-
tion.
The difficulty is that the number of possible journeys increases enormously.
A formulation is required that summarizes the effects of the availability of an
infinite number of travel alternatives without enumerating,
6.1.2
Subjective Utility Ranking
In the present model, the number of people in a class that choose a journey
depends on the journey's costs and on the desirability ranking of all the journeys.
The utility values of the journeys are not considered, except to determine the
ranking.
53
This is reasonable when journeys are not similar.,
However, when they
are nearly the same, we can expect that different people will have different
opinions on relative rankings.
Therefore, the model should be changed to renre-
sent the effects of having journeys with nearly equal utilities.
to incorporate into the density function fa(...)
One approach is
That
an indication of ranking.
is, if there are k journeys available to class a let
T
be a permutation of
{l,...,k}, which represents a possible ranking of the journeys.
Then let
D fa (u,n,T))du dn,
be the number of people in class a with time and money costs between u and
u + du and n and n + dn, respectively, and who have preference order I.
analysis that has led to equations (2.20) to
possible preference order.
The
(2.22) must be repeated for each
If, as we suspect, there are not many likely
permutations, the increase in complexity that this causes will be limited
and manageable.
6.1.3
Probalistic Preference Modeling
It may be argued that preferences are almost never as clear cut as we
have modeled them here.
That is, there is rarely a universally agreed preference
order among travel alternatives, even among those of the same socio-economic
class.
In that case, a better behavior principle may be the following.
and path i, let
a known way.
aibe a probability that depends on W.,
For example,
ia may be proportional to Wi.
For class a
the utility of path i, in
Let the probability
that a given traveler of class a who can afford to take path i be equal to T .
6.1.4
Correlations Within Households
We have modeled all individuals as being completely independent.
members of a household share resources and responsibilities.
In fact,
For example, they
share income so a reduction in one member's expenditure may lead to an increase
in another's budget.
In addition, if one member goes to a supermarket on a given
day, others are less likely to go to a supermarket.
On the other hand, if one
54
I
member goes to a movie, others are more likely to.
Incorporating this correlation may have an important effect on network
results.
For example, as pointed out above, reducing bus fares may increase car
travel, since the amount of bus travel is likely to be unchanged and more money is
available for car travelers.
6.1.5
Residential Choice
We have considered only some of the choices available to travelers that affect
their travel behavior.
In particular, we have not allowed them the freedom to
choose their residences; we assume that their residential locations are fixed.
A
more realistic and comprehensive model would include a representation of residential
choice.
The choice of residential location is greatly influenced by the transportation system:
when a household considers a new residence, it carefully
considers how much access it has to locations of work, shopping, and where
other needs are fulfilled.
as housing cost.
Other factors must be considered as well, such
Housing cost itself is determined in part by these same
transportation considerations.
It is desirable to include residential choice in the model presented here to assess long-term effects of changes in the transportation system.
To do this, the following modifications of the method must
be considered:
a.
Classes no longer have fixed origins but rather can choose journeys
anywhere in the network.
Some of the nodes on each resulting journey
become residential locations.
b.
A precise model of individual behavior incorporating both residential
and travel decisions is required.
to (2.9).
This may be an extension of
(2.7)
It is hoped that the utility can be extended to include
the intrinsic utility (i.e., independent of transportation considerations) of a house or apartment, and that the money budget can be
enlarged to include housing as well as travel costs.
55
It is also
hoped that such a model can be transformed into a principle that
describes flows and demands for housing which is analogous to (2.20)
to (2.22).
c.
A model of housing cost is required.
The cost of owning or renting
a house or apartment depends on its physical
characteristics (such
as its size, the size of the plot of land it is on, its state of
repair), and on market
factors such as interest rates.
It also
depends on the demand for residences in its vicinity which in turn
depends on transportation considerations.
These modifications will
Travelers will be represented as
extend the model described here.
making both travel decisions and residence decisions.
Both phenomena
must be considered simultaneously if long-term changes in urban structure
are to be understood.
5.2
Analytical Research
A major limitation to the new assignment principle is in its computational
complexity and large data base requirements when large networks are treated.
Most important, as the network grows, the number of journeys (possible journeys
The lengths of these journeys, measured
as well as those actually used) increases,
in the number nodes passed through, may also grow.
the journeys were all generated manually,
In the example reported here.
In the large network, simple loops
were generated by hand and the computer examined all concotenations to form journeys.
efficient exact or heuristic method is required to generate journeys in large
A.,n
networks.
The computational difficulties are exacerbated by some of the extensions
described above.
Further research to accelerate the equilibration process, or to
replace it altogether, will help mitigate these difficulties.
Of greater value
will be a procedure for aggregating the network so that a reduced network can be
used which will produce approximately equivalent results.
There are important qualitative issues that must be understood, such as the
number of equilibria.
We have briefly touched on the effect of the shape of the
budget distributions.
Other sensitivity issues, such as the effect of the shape
of the link delay functions should be considered.
56
If the full benefits of symmetry are used, very large networks can be treated,
particularly concentric circle networks, such as in Figure 4.7.
Although such
studies will have no direct application, they will be useful for the study of
the behavior of travelers in large abstract city networks,
That is, general
principles for understanding transportation systems can be developed by considering such special cases.
Optimization techniques based on this model are required.
Such techniques
will seek the best of a class of network modifications or the best of a class
of control policies.
They will minimize such costs as total funds expended, ag-
gregate delay, energy consumption, or maximize such objectives as mobility
while taking into account travelers' responses to the change,
These optimization
techniques will be of great value to system planners and managers.
6.3
Empirical Research
Various assumptions have been made in formulating the examples,
Most
important may be the assumption that utility is additive along journeys.
Such
assumptions must be tested and verified or modified.
A study can be undertaken to apply the methods described here to a real
city.
The city, of course, should be small and well-documented in order to assess
the predictive power of the model.
57
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59